Radius/interval of convergence

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Homework Help Overview

The discussion revolves around determining the radius and interval of convergence for the series \(\sum_{n=1}^{\infty} n!(2x-1)^n\). Participants are analyzing the application of the ratio test in this context.

Discussion Character

  • Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to apply the ratio test but expresses uncertainty about the correctness of their results, particularly regarding the limits involved. Some participants question the handling of the limit as \(n\) approaches infinity, specifically the treatment of \(|n+1|\).

Discussion Status

Participants are exploring different interpretations of the limit and its implications for convergence. There is a recognition of a specific value of \(x\) that may lead to convergence, but no consensus has been reached on the overall interval or radius of convergence.

Contextual Notes

There is mention of a discrepancy between the original poster's findings and the expected results from a reference, which may indicate a misunderstanding or miscalculation in the application of the ratio test.

karadda
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Homework Statement

[tex]\Sigma[/tex] n!(2x-1)[tex]^{}n[/tex]

from n=1 to infinity

Homework Equations



-ratio test

The Attempt at a Solution



lim n-> infinity | (n+1)!(2x-1)^(n+1) / n!(2x-1)^n |

lim n-> infinity | (n+1)(2x-1) |

|2x - 1| lim n-> infinity | (n+1) |

-1 < 2x -1 < 1

0 < 2x < 2
0 < x < 1
I know at this point I've done something pretty wrong. as my interval and radius doesn't match up with the back of the book. could use a push in the right direction, thanks.
 
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What happened to the lim n->infinity |n+1|? Did you just drop it?
 
oh, I am not too sure what to do with it. i understand that the limit goes to infinity, UNLESS x = 1/2 in which case it goes to 0. how do i proceed knowing that?

nm, i should just stop there ;)

1/2 is the only value of x for which this will converge, so the interval of convergence is {1/2}. the radius is 0.
 
Last edited:
karadda said:
oh, I am not too sure what to do with it. i understand that the limit goes to infinity, UNLESS x = 1/2 in which case it goes to 0. how do i proceed knowing that?

nm, i should just stop there ;)

1/2 is the only value of x for which this will converge, so the interval of convergence is {1/2}. the radius is 0.

I couldn't have said it better myself.
 

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